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Use the given information to find the unknown value:
y varies directly as the cube of
x. When
x=2, then
y=16. Find
y when
x=3.

Use the given information to find the unknown value: $y$ varies directly as the cube of $x$ . When $x=2$ , then $y=16$ . Find $y$ when $x=3$ .

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Write and solve direct variation equations

Full solution

Q. Use the given information to find the unknown value:

y

varies directly as the cube of

x

. When

x=2

, then

y=16

. Find

y

when

x=3

.

Establish Relationship: Establish the direct variation relationship between $y$ and the cube of $x$ . Since $y$ varies directly as the cube of $x$ , we can write the equation as $y = kx^3$ , where $k$ is the constant of proportionality.
Find Constant of Proportionality: Use the given values to find the constant of proportionality $k$ . We know that $y = 16$ when $x = 2$ . Substitute these values into the equation $y = kx^3$ to find $k$ . $16 = k \times 2^3$ $16 = k \times 8$
Solve for k: Solve for k. $\newline$ Divide both sides by $8$ to isolate k. $\newline$ $k = \frac{16}{8}$ $\newline$ $k = 2$
Write Equation with $k$ : Write the direct variation equation with the found value of $k$ . Now that we know $k = 2$ , we can write the equation as $y = 2x^3$ .
Find $y$ for $x=3$ : Find $y$ when $x = 3$ using the direct variation equation. $\newline$ Substitute $x = 3$ into the equation $y = 2x^3$ to find $y$ . $\newline$ $y = 2 \times 3^3$ $\newline$ $y = 2 \times 27$ $\newline$ $y = 54$

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